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MATHEMATICS
A. S. FOKHT
SOME ESTIMATES NEAR THE BOUNDARY OF A DOMAIN FOR A POLYHARMONIC FUNCTION AND ITS DERIVATIVES DEFINED IN AN \(N\)-DIMENSIONAL DOMAIN
(Presented by Academician A. I. Mal’tsev on 16 VI 1962)
In the present paper, estimates will be obtained for a polyharmonic function and its partial derivatives of arbitrary order in the \(L_2\) metric in an arbitrary \(N\)-dimensional domain \(G\) with boundary \(\Gamma\) of class \(C^{(2)}\). The investigations are based on the estimates obtained in the papers \((^{1,2})\) and are carried out essentially in the three-dimensional case. They are transferred to the \(N\)-dimensional case by analogy.
§ 1. Let \(N=3\) and
\[ \Delta^l u=0, \tag{1,1} \]
\(l>0\) an integer, and let
\[ I=\int_0^R \int_0^{2\pi} \int_0^\pi u^2 r^2 \sin\theta\, dr\, d\theta\, d\varphi < +\infty; \tag{1,2} \]
\[ I_r^{(q)}=\int_0^{2\pi}\int_0^\pi u^{(q)2} r^2 \sin\theta\, d\theta\, d\varphi, \tag{1,3} \]
where \(u^{(q)}\) is any mixed derivative taken of the function \(u\) with respect to the variables \(r,\theta,\varphi\); \(q>0\) is an integer.
In the proposed paper, first, the inequality will be proved
\[ I_r^{(q)} \leqslant \frac{C_{l,q}}{(R-r)^{2q+1}}\, I. \tag{1,4} \]
Next it is proved that if \(u\) is an \(l\)-harmonic function defined in a three-dimensional domain \(G\) with boundary \(\Gamma \in C^{(2)}\), then there exists a piecewise smooth (consisting of pieces of spheres) closed (geometric) surface \(\gamma_\delta \subset G\), possessing the following properties:
1) The distance \(\rho(\bar{x},\Gamma)\) from each point \(\bar{x}\in\gamma_\delta\) to \(\Gamma\) satisfies the inequalities
\[ {}^{1}/_{3}\delta \leqslant \rho(\bar{x},\Gamma)\leqslant {}^{1}/_{2}\delta \quad (0<\delta<\delta_0). \tag{1,5} \]
2) For each point \(P\in\Gamma\) there exists a unique point \(Q\in\gamma_\delta\) lying on the inner normal to \(\Gamma\) drawn from \(P\), and \(\gamma_\delta\) contains no other points except
\[ Q=\varphi(P). \tag{1,6} \]
3) The inequality holds
\[ \mu_1\, d\Gamma \leqslant d\gamma_\delta \leqslant \mu_2\, d\Gamma \tag{1,7} \]
(\(\mu_1,\mu_2\) are positive constants), where \(d\gamma_\delta\) is the differential near the point \(Q\) of any smooth piece of the surface \(\gamma_\delta\); \(d\Gamma\) is the differential of the surface near the point \(P\).
surface near the point \(P\), corresponding to \(Q\) (see formula (1.6)). In this case the inequality
\[ I_{\gamma_\delta}^{(q)} \leqslant \frac{C_{l,q}^{*}}{\delta^{2q+1}}\, I_0, \tag{1.8} \]
holds, where
\[ I_{\gamma_\delta}^{(q)}=\iint_{\gamma_\delta} u^{(q)2}\,d\gamma_\delta; \tag{1.9} \]
\[ I_0=\iiint_G u^2\,dG. \tag{1.10} \]
§ 2. The proof of estimates of the form (1.4) in the case when the \(q\)-th derivatives are computed with respect to the variables \(r\) and \(\varphi\) reduces directly to inequalities proved in paper (1). What remains to be considered is the proof concerning derivatives with respect to the variable \(\theta\).
Lemma. Let \(f(r)\) be a positive polynomial of degree \(4(l-1)\). Then for all natural \(n \geqslant 0\) the inequality
\[ \int_0^1 r^{2n+2} f(r)\,dr \geqslant \varkappa_l \int_0^1 r^{2n+1} f(r)\,dr, \tag{2.1} \]
holds, where \(\varkappa_l\) is a positive constant depending only on \(l\).
Let a harmonic function be given in the unit ball \((N=3,\ 0\leqslant r<1)\):
\[ v=\sum_{n=0}^{\infty} r^n \sum_{k=0}^{n} A_{nk}\cos k\varphi \cdot P_n^{(k)}(\cos\theta). \]
By virtue of the work of S. M. Nikol’skii \((^2)\), we shall have
\[ \int_0^{2\pi}\int_0^\pi \left\{ \sum_{n=0}^{\infty} r^n \sum_{k=0}^{n} A_{nk}\cos k\varphi \left[P_n^{(k)}(\cos\theta)\right]_{\theta}^{(q)} \right\}^{2} r\sin\theta\,d\theta\,d\varphi \leqslant \]
\[ \leqslant \frac{C_q}{(1-r)^{2q+1}} \sum_{k=0}^{\infty}\sum_{n=k}^{\infty} \frac{A_{nk}^{2}(n+k)!}{(n+1)^2(n-k)!}. \tag{2.2} \]
Let us note that inequality (2.2) is valid for arbitrary numbers \(A_{nk}\) independent of \(\varphi\) and \(\theta\), and for any \(r\) \((0\leqslant r<1)\).
Put
\[ A_{nk}^{2}=F_{nk}= \left[ a_{1n}^{(k)}+a_{2n}^{(k)}r^2+\cdots+a_{ln}^{(k)}r^{2(l-1)} \right]^2, \tag{2.3} \]
where \(a_{in}^{(k)}\) \((i=1,2,\ldots,l)\) are linear combinations of the Fourier coefficients of the boundary values of the \(l\)-harmonic function \(u\) on the unit ball.
Next consider
\[ \Phi_{nk}= \int_0^1 r^{2n+1} \left[ a_{1n}^{(k)}+a_{2n}^{(k)}r^2+\cdots+a_{ln}^{(k)}r^{2(l-1)} \right]^2\,dr, \tag{2.4} \]
\[ \Phi_{nk}^{*}= \int_0^1 r^{2n+2} \left[ a_{1n}^{(k)}+a_{2n}^{(k)}r^2+\cdots+a_{ln}^{(k)}r^{2(l-1)} \right]^2\,dr. \tag{2.5} \]
By virtue of the lemma we have
\[ \Phi_{nk}<\varkappa_l\Phi_{nk}^{*}. \tag{2.6} \]
In paper (¹) the relation
\[ |F_{nk}| \leq C_l \Phi_{nk}(n+1), \tag{2,7} \]
valid for all \(n\) and \(k\) (\(C_l>0\) is a constant), was proved.
By virtue of (2,2), (2,3), (2,6), (2,7), and the known representation of a harmonic function in a three-dimensional ball through \(a_{ln}^{(k)}\), we shall have
\[ \begin{aligned} I_r^{(q)} &= \int_0^{2\pi}\int_0^\pi \left\{ \sum_{n=0}^{\infty} r^n \sum_{k=0}^{n} \sqrt{|F_{nk}|}\cos k\varphi \left[P_n^{(k)}(\cos\theta)\right]_{\theta}^{(q)} \right\}^{2} r\sin\theta\,d\theta\,d\varphi \leq \\ &\leq \frac{C_q}{(1-r)^{2q+1}} \sum_{k=0}^{\infty}\sum_{n=k}^{\infty} \frac{F_{nk}(n+k)!}{(n+1)^2(n-k)!} \leq \frac{C_l C_q}{(1-r)^{2q+1}} \sum_{k=0}^{\infty}\sum_{n=k}^{\infty} \frac{\Phi_{nk}(n+1)!}{(n+1)(n-k)!} < \\ &< \frac{C_{l,q}}{(1-r)^{2q+1}} \sum_{k=0}^{\infty}\sum_{n=k}^{\infty} \frac{\Phi_{nk}^{*}(n+k)!}{(n+1)(n-k)!} = \frac{C_{l,q}}{(1-r)^{2q+1}}\,I , \end{aligned} \]
where \(C_{l,q}=C_l C_q \chi_l\), which was to be proved.
In the case \(N>3\) the proof is analogous.
§ 3. For definiteness, let us take \(N=3\); however, the method used is general for any \(N\). Consider an arbitrary bounded three-dimensional domain \(G\) with boundary \(\Gamma\) of class \(C^{(2)}\).
Taking as a basis the results of the work of S. M. Nikol’skii (³), near each point \(x_0=(x_1^0,x_2^0,x_3^0)\in\Gamma\) we construct a neighborhood \(\Delta=\Delta(\bar x_0)\) which cuts out on \(\Gamma\) a piece \(\sigma\) defined by an equation explicitly expressed through one of the coordinates. For definiteness let this equation be
\[ x_3=\varphi(x_1,x_2),\qquad \Delta=\{a<x_1<b;\ c<x_2<d\},\qquad \varphi\in C^{(2)}(\bar\Delta). \tag{3,1} \]
Put
\[ u_1=x_1+h\alpha_1,\qquad u_2=x_2+h\alpha_2,\qquad u_3=\varphi(x_1,x_2)+h\alpha_3, \tag{3,2} \]
where \(h>0\); \(\alpha_i=\alpha_i(x_1,x_2)\) are the direction cosines of the angles formed by the inner normal to \(\Gamma\) with the axes \(x_i\). For sufficiently small \(\delta_0>0\), the equalities (3,2) give a one-to-one and continuously differentiable transformation of the points \((x_1,x_2,h)\) \(((x_1,x_2)\in\Delta,\ 0<h<\delta_0)\) into the points \((u_1,u_2,u_3)\in U_{\Delta,\delta_0}\subset G\). Define also the domain
\[ U_{\Delta',\,\delta_0/2}\subset U_{\Delta,\delta_0},\qquad \Delta'=\left\{a+\frac{\delta_0}{2}<x_1<b-\frac{\delta_0}{2};\ c+\frac{\delta_0}{2}<x_2<d-\frac{\delta_0}{2}\right\}. \]
Let
\[ x_i=\psi_i(u_1,u_2,u_3),\qquad h=\psi_3(u_1,u_2,u_3)\quad (i=1,2) \tag{3,3} \]
be the transformations inverse to (3,2), and
\[ \left|\frac{\partial\psi_k}{\partial u_j}\right|\leq M \qquad (k,j=1,2,3),\qquad (u_1,u_2,u_3)\in U_{\Delta,\delta_0}. \]
Then, according to (³), any ball \(\omega\) with center \(Q(x_1^0,x_2^0,\delta_0/2)\in U_{\Delta',\delta_0/2}\) and radius \(\mu\delta_0/2\) belongs to the domain \(U_{\Delta,\delta_0}\), i.e. does not go beyond the boundary \(\Gamma\) of the domain \(G\), provided only that \(0<\mu\leq 1/3M\). By the properties of the function \(\varphi(x_1,x_2)\), \(|\partial\varphi/\partial l|\leq L<+\infty\), where \(l\) is any direction in the plane \((x_1,x_2)\). Take \(T=\max(L,M)\).
Construct on \(\Delta'\) a grid of squares with side \(h=\delta_0/96(1+T)^3\), and a grid of concentric balls \(V\) and \(V_1\) of radii \(\rho=7\delta_0/48(1+T)\) and \(\rho_1=\delta_0/8(1+T)\), respectively, with centers lying on the surface \(\Gamma_{\delta_0/2}\) (\(\Gamma_\eta\) is the surface at distance \(\eta\) from \(\Gamma\) along the inner normals) and projecting in the direction \(h\) to the nodes of the grid on \(\Delta'\).
On the basis of the Heine–Borel lemma, the surface $\Gamma$ can be covered by a finite number of the neighborhoods $\Lambda(\bar{x}_0)$ defined above, and in each such neighborhood one can construct the above-described net of balls $V$ and $V_1$.
Obviously, there exists a closed (geometrically) piecewise-smooth surface $\gamma_\delta \subset G$ satisfying conditions (1,5) and (1,6). In addition, in the present paper it has been shown that the balls $V_1$ (of smaller radius) completely contain the volume layer determined by the inequality
\[ \frac{\delta_0}{2}-\frac{\delta_0}{16(1+T)} \leq h \leq \frac{\delta_0}{2}, \]
whence inequality (1,7) follows. Let us also note that each ball $V$ of the construction intersects only a finite number of balls.
Using inequality (1.4), we obtain
\[ \iint_{\gamma_\delta} u^{(q)2}\,d\gamma_\delta < \sum_{\nu=1}^{n} \iint_{C_{r_\nu}} u^{(q)2}\,dS < \frac{C_{l,q}}{\delta^{2q+1}} \sum_{\nu=1}^{n} \iiint_{V_\nu} u^2\,dV_\nu < \frac{C^{*}_{l,q}}{\delta^{2q+1}} \iiint_G u^2\,dG, \]
which is what was required to prove. Here $C^{*}_{l,q}=mC_{l,q}$, where $m$ is the greatest number of balls $V$ of the net that intersect any fixed ball of the net; the number $m$ depends on $G$, but not on $\delta_0$; $n$ is the total number of balls $V$ of the net, $\nu$ is the serial number of the ball $V_\nu$; $C_{r_\nu}$ is the full surface of the ball $V_{1\nu}$.
The estimates obtained, (1,4) and (1,8), may be used in the solution of certain boundary-value problems, for example, in proving uniqueness of the solution of the first boundary-value problem for a polyharmonic equation.
Moscow Institute of Physics and Technology
Received
31 V 1962
REFERENCES
- A. S. Foht, DAN, 147, No. 1 (1962).
- S. M. Nikol’skii, Siberian Math. Journal, 1, No. 1 (1960).
- S. M. Nikol’skii, Izv. AN SSSR, Ser. Math., 22, No. 5 (1958).